Numerical evaluation at negative integers of the Dedekind zeta functions of totally real cubic number fields

Abstract

Let K be a totally real number field. It is known that the values ζK(-n) of the Dedekind zeta function ζK(s) of K are rational numbers for all non-negative integers n ≥ 1. We develop a rigorous and reasonably fast method for computing these exact values. Our method is in fact developed in the case of totally real number fields K of any degree for which ζK(s)/ζ(s) is entire, which is conjecturally always the case (and holds true if K is cubic or if K/Q is normal). © Springer-Verlag Berlin Heidelberg 2004.